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Lesson M19.L01: Beyond Solow: Why Countries Differ in Income Levels

Module: Economic Growth Part II Level: intermediate Duration: 30 minutes Learning Objective: Identify the limitations of the Solow model in explaining cross-country income differences. Data as of: 2024 Provenance: Productivity Commission | World Bank Open Data

Explanation

The Solow growth model predicts conditional convergence: countries with the same savings rate s, depreciation rate δ, and technology level A will converge to the same steady-state capital per worker k* and hence the same income per worker y*. Countries that start with lower k grow faster until they reach *k*. This is an elegant result, but how well does it fit the data?

The 100:1 problem. Income per capita in the United States is roughly 100 times that of the world's poorest countries (e.g., Liberia, Malawi). Pure Solow explains at most 30–40% of the variance in cross-country incomes. Even allowing for wide differences in s and δ, the model cannot generate income gaps this large without implausibly large technology gaps.

What Solow misses: - Human capital (skills, education, health). Workers in rich countries are far more productive per hour. - Institutions (property rights, rule of law, contract enforcement). - Geography and disease burden. - Technology gaps — Solow treats A as exogenous and identical across countries in the long run.

Augmented Solow (Mankiw, Romer & Weil 1992). Adding human capital H to the production function — Y = K^α H^β (AL)^(1−α−β) — raises the model's explanatory power to roughly 80% of cross-country income variance. But technology (A) remains exogenous; the model still cannot explain why A differs.

Australian context. Australia's GDP per capita (~USD 65,000) is close to the OECD average and about 20 times that of Papua New Guinea (~USD 3,200). Pacific island nations share similar geographies but very different incomes. The Solow framework attributes this gap to differences in k and A, but cannot say where the A differences come from — motivating Modules M19.L03 and M19.L04.

Notation summary: - Y = output; K = physical capital; L = labour; A = technology (total factor productivity) - s = savings rate; δ = depreciation rate; k* = steady-state capital per worker - α, β = output elasticities of capital and human capital

Worked Example

Question: Suppose two countries have identical technology A and depreciation δ = 0.05, but Country A saves 30% of income (s = 0.30) and Country B saves 10% (s = 0.10). In the basic Solow model with production function Y = K^0.5 (AL)^0.5 (so α = 0.5), how large is the difference in steady-state income per effective worker?

Step 1 — Steady-state condition.

At steady state, investment = depreciation:

\[s \cdot y^* = \delta \cdot k^*\]

Since y = k^α (in per-effective-worker terms with A = 1 for simplicity):

\[s \cdot (k^*)^{0.5} = \delta \cdot k^*\]

Step 2 — Solve for k*.

\[s = \delta \cdot (k^*)^{0.5}\]
\[(k^*)^{0.5} = \frac{s}{\delta}\]
\[k^* = \left(\frac{s}{\delta}\right)^2\]

Step 3 — Steady-state output per worker.

\[y^* = (k^*)^{0.5} = \frac{s}{\delta}\]

Step 4 — Compare the two countries.

Country A: y*_A = 0.30 / 0.05 = 6

Country B: y*_B = 0.10 / 0.05 = 2

\[\frac{y^*_A}{y^*_B} = \frac{6}{2} = 3\]

Step 5 — Interpret.

A threefold saving rate difference (30% vs 10%) produces only a 3:1 income ratio at steady state. But observed income ratios reach 100:1. The basic Solow model therefore cannot account for observed cross-country income differences from saving-rate differences alone — even large differences in s generate modest income gaps when α = 0.5.

Common Misconception

Misconception: "The Solow model predicts that all countries will eventually converge to the same income level."

Correction: Solow predicts conditional convergence, not absolute convergence. Countries converge to their own steady state, which depends on their individual parameters (s, δ, n, A). Countries with different structural parameters can have permanently different steady-state incomes. Absolute convergence (all countries to the same level) would only occur if every country had identical parameters — an assumption contradicted by institutions, geography, and policy differences.

Practice Prompts

  1. Conceptual: Why does the augmented Solow model (Mankiw, Romer & Weil 1992) explain a larger share of cross-country income variance than the basic Solow model?

→ Answer: The augmented model adds human capital H as a separate factor. Countries differ not just in physical capital K but also in education and skills. Including H captures an additional dimension of capital accumulation that the basic model attributes entirely to unexplained TFP differences, thereby raising R² from ~30–40% to ~80%.

  1. Numerical: In the Solow model with α = 1/3, s = 0.24, δ = 0.06, and A = 1 (setting n = g = 0 for simplicity), the steady-state capital per worker is k* = (s/δ)^(1/(1−α)). Calculate k* and y* = (k*)^(1/3).

→ Answer: - 1/(1−α) = 1/(1−1/3) = 1/(2/3) = 3/2 - s/δ = 0.24/0.06 = 4 - k* = 4^(3/2) = (4^1)^(3/2) = 4 × √4 = 4 × 2 = 8 - y* = 8^(1/3) = 2

So steady-state output per worker is 2 units.

  1. Application: Australia and Papua New Guinea are geographically close but have income per capita ratios of roughly 20:1. Using the Solow framework, what structural factors might explain this gap? Which factors does the basic Solow model capture, and which does it miss?

→ Answer: The basic Solow model can attribute the gap to differences in the savings rate s and technology A. Australia has a higher household savings rate, deeper capital markets, and more advanced technology. However, a 20:1 gap is too large to be explained by s differences alone (as the worked example shows). Solow misses institutional quality (property rights, rule of law), human capital (education, health), and the sources of technology differences — all of which are substantially higher in Australia than PNG.

Further Resources